A generalized PGL(2) Petersson/Bruggeman-Kuznetsov formula for analytic applications

Abstract

We develop generalized Petersson/Bruggeman-Kuznetsov (PBK) formulas for specified local components at non-archimedean places. In fact, we introduce two hypotheses on non-archimedean test function pairs $f\leftrightarrow \ddot{I}€(f)$, called geometric and spectral hypotheses, under which one obtains `nice' PBK formulas by the adelic relative trace function approach. Then, given a supercuspidal representation $\ddot{I}ƒ$ of ${\mathrm{PGL}}_2({\mathbb{Q}}_p)$, we study extensively the case that $\ddot{I}€(f)$ is a projection onto the line of the newform if $\ddot{I}€$ is isomorphc to $\ddot{I}ƒ$ or its unramified quadratic twist, and $\ddot{I}€(f)=0$ otherwise. As a first application, we prove an optimal large sieve inequality for families of automorphic representations that arise in our framework.

v3: Final accepted version. To appear in Forum Math. Sigma